Some first birthday parties include homemade birthday cakes and photos of babies with frosting all over their faces – but few babies have Martha Stewart for a grandmother.

Last Thursday, Stewart and her daughter Alexis hosted an intimate celebration at Perry St., a restaurant in New York, to toast Alexis's daughter Jude's first birthday.

Catered by legendary chef Jean-Georges Vongerichten and his son, Cedric, who is also the executive chef at the eatery, the pièce de résistance was a trio of homemade birthday cakes in a variety of flavors topped with marzipan animals.

Martha Stewart certainly is not offended by the fact her daughter, Alexis Stewart, is unleashing a tell-all about growing up under the heavy thumb of her mother.

Not only is Stewart's Blackberry buzzing about the book – "I get a little 'ping ping' like every five minutes," says the domestic diva, 70 – but, she says, "I got an advanced copy and I actually read the book … oh, maybe about a month or a month and a half ago. I didn't know that it was even coming."

Whateverland: Learning to Live Here, which Alexis, 46, co-wrote with her Whatever Sirius Satellite Radio-show partner Jennifer Koppelman Hutt, is to be published Oct. 18 and contains "fabulous pictures," says Martha. "And it is hilarious and it is enlightening and it's full of funny stories."

Was über-hostess and domestic darling Martha Stewart more like Mommie Dearest as a parent?

Her daughter Alexis, 46, levies some harsh claims in her new book, Whateverland: Learning to Live Here, co-authored with Jennifer Koppelman Hutt.

"Martha does everything better! You can't win," Alexis Stewart writes of her 70-year-old mom, according to an excerpt obtained by U.K.'s Daily Mail. "If I didn't do something perfectly, I had to do it again. … I grew up with a glue gun pointed at my head."

As for holidays, Alexis – who recently welcomed a daughter of her own, Jude – writes that special moments like Halloween and Christmas held little excitement for her mother.

Martha Stewart's only child, Alexis, has welcomed a daughter of her own.

The baby girl, Jude, was born via gestational carrier on Tuesday.

Though the entertaining guru is surely glad to have a granddaughter, don't expect her to be a typical grandmother. "Martha is going to be called, 'Martha,'" Alexis announced on her radio show, Whatever with Alexis & Jennifer. "She seemed to have no problem with it. ... I was like, 'Do you really want to be 'grandma'? I don't like the term grandma!"

For the past few years, Alexis Stewart, 45, has been open about her struggle to conceive a child, telling PEOPLE in 2007 that she was spending more than $20,000 each month on fertility treatments.